The $B_2$ Harmonic Oscillator with Reflections and Superintegrability

The two-dimensional quantum harmonic oscillator is modified with reflection terms associated with the action of the Coxeter group $B_2$, which is the symmetry group of the square. The angular momentum operator is also modified with reflections. The wavefunctions are known to be built up from Jacobi and Laguerre polynomials. This paper introduces a fourth-order differential-difference operator commuting with the Hamiltonian but not with the angular momentum operator; a specific instance of superintegrability. The action of the operator on the usual orthogonal basis of wavefunctions is explicitly described. The wavefunctions are classified according to the representations of the group: four of degree one and one of degree two. The identity representation encompasses the wavefunctions invariant under the group. The paper begins with a short discussion of the modified Hamiltonians associated to finite reflection groups, and related raising and lowering operators. In particular, the Hamiltonian for the symmetric groups describes the Calogero-Sutherland model of identical particles on the line with harmonic confinement.


Introduction
The two-dimensional quantum harmonic oscillator is modified with reflection terms associated with the action of the Coxeter group B 2 , the symmetry group of the square.The wavefunctions are known to be built up from Jacobi and Laguerre polynomials.This paper introduces a fourthorder differential-difference operator commuting with the Hamiltonian but not with the angular momentum operator; a specific instance of superintegrability.The action of the operator on an orthogonal basis of wavefunctions is explicitly described.The wavefunctions are not in general invariant under the group, rather are classified by the representations of the group: four of degree one and one of degree two.The group-invariant wavefunctions of the B 2 oscillator and its superintegrability have been studied by Tremblay et al. [10,11], Quesne [9].
First the general background on finite reflection groups and root systems, Dunkl operators, and the associated Hamiltonian is described.In particular, the Calogero-Sutherland model of N identical particles on a line with r −2 interaction and harmonic confinement comes from the symmetric group (Lassalle [8], Baker and Forrester [1]).In the general situation, there are raising and lowering operators which can be used to construct operators commuting with the Hamiltonian and the group action.After this the development turns to dihedral groups (type I 2 (k)) and the use of a complex coordinate system, which simplifies the description of rotations.Some general formulas are specialized to this setting.
The description of the wavefunctions of B 2 and of the action of specific operators is in Sections 4 and 6.The important operators are the Hamiltonian H, the angular momentum J and a new operator K which commutes with H and the group action but not with J 2 .This property constitutes superintegrability.There are a number of different classes of wavefunctions, requiring frequent case-by-case analysis.The explicit action of K on the orthogonal basis of wavefunctions is found in Section 7.
In the appendix, there are details on some proofs, and a sketch of a symbolic computation method of proving relations involving polynomials and Dunkl operators.

Reflection groups and a harmonic oscillator
In R N the inner product is ⟨x, y⟩ := N i=1 x i y i and ∥x∥ 2 = ⟨x, x⟩.If v ̸ = 0, then the reflection σ v along v is defined by This is an isometry ∥xσ v ∥ 2 = ∥x∥ 2 and an involution σ 2 v = I.The set of fixed points (xσ v = x) is the hyperplane {x : ⟨x, v⟩ = 0}.A finite root system is a subset R of nonzero elements of R N satisfying u, v ∈ R implies uσ v ∈ R. We restrict consideration to reduced root systems, that is if u, cu ∈ R, then c = ±1.Define W (R) to be the group generated by {σ v : v ∈ R}; this is a finite subgroup of the orthogonal group O N (R).There is a decomposition of R into R + (the positive roots) and R − ; this relies on choice of a vector u such that ⟨u, v⟩ ̸ = 0 for all v ∈ R then set R + = {v ∈ R : ⟨u, v⟩ > 0}.Since σ v = σ −v , the set R + can be used to index the reflections in W (R). The set of reflections σ v decomposes into conjugacy classes (W orbits) σ u ∼ σ v if u = vw for some w ∈ W (R). A multiplicity function κ v is a function on R which is constant on each conjugacy class, usually here κ v ≥ 1. Set γ κ := v∈R + κ v .Define the Dunkl operator Then D i D j = D j D i for all i, j (Dunkl [2], also see Dunkl and Xu [4,Theorem 6.4.8]).Let This leads to the modified Schrödinger equation (with parameter ω > 0) The exponential ground state is g(x) := exp − ω 2 ∥x∥ 2 , as can be seen from the transformation (2.1) Denote the set of polynomials on R N by P and the set of polynomials homogeneous of degree n by P n (that is, p(cx) = c n p(x) for c ∈ R).Let H κ,n = {p ∈ P n : ∆ κ p = 0} (κ-harmonic polynomials).We find eigenfunctions of g −1 Hg of the form p(x)q ω∥x∥ 2 with p ∈ H κ,n (thus ∆ κ p = 0 and ⟨x, ∇⟩p = np).This gives the differential equation (where t = ω∥x∥ 2 ) and the solution is the Laguerre polynomial q(t) = L (α) ).Note E depends on deg(pq) = n + 2m.The Laguerre polynomial of degree n and index α > −1 satisfies The Pochhammer symbol is (a There is an orthogonality structure which uses the W (R)-invariant weight function positively homogeneous of degree γ κ .The orthogonality H κ,n ⊥H κ,m for n ̸ = m holds with respect to the measure h κ (x) 2 dµ(x) on the sphere S N −1 := {x : ∥x∥ = 1}, where µ is the rotation-invariant surface measure.There is a key result on adjoints: suppose p, q are sufficiently smooth and have exponential decay then (with where dm is Lebesgue measure on R N (see [4,Theorem 7.7.10]).Thus the adjoint of D i is defined on a dense subspace of L 2 R N , h 2 κ dm and D * i = −D i .This meaning of adjoint will be used throughout.Furthermore, the conjugate of H is (details of the derivation are in Appendix A) a Schrödinger equation with the potential which includes reflections.The ground state is h κ g.For the special case where R is the root system of type A N −1 and W (R) = S N (the symmetric group), this potential occurs in the Calogero-Sutherland model of N identical particles on a line with r −2 interaction potential and harmonic confinement.There is a closely related model of N identical particles on a circle with r −2 interaction, called the trigonometric model.The wavefunctions are Jack polynomials in the variables x j = e iθ j , 1 ≤ j ≤ N .Lapointe and Vinet [7] defined raising and lowering operators and found Rodrigues formulas for the Jack polynomials arising in this model.The Jack polynomials can be used as bases for generalized Hermite (Lassalle [8]) and Laguerre polynomials, which occur as wavefunctions in types A and B models on the line (Baker and Forrester [1], also see [4,Section 11.6.3]).We need the basic commutation relations ([A, B] :  This family of angular momentum operators has been studied by Feigin and Hakobyan [6], especially in connection with the symmetric group and the Calogero-Moser model. We introduce raising and lowering operators.These operators were used by Feigin [5] in his study of generalized Calogero-Moser models, which are constructed in terms of subdiagrams (certain subsets of roots) of the Coxeter diagram of W (R). Note {A, B} := AB + BA.
Proof .From (2.2), it follows that (A + a ) * = A − a and H * a = H a .The commutator and expanding the right hand side with formulas (2.4) and Proof .This follows from ⟨a, ∇ κ ⟩w = w⟨aw, ∇ κ ⟩ (see [4,Proposition 6.4.3]) and This produces a collection of self-adjoint operators commuting with W (R) and H.

The dihedral groups
For m = 3, 4, . .., the dihedral group I 2 (m) is the symmetry group of the regular m-gon.We will use complex coordinates for R 2 : Let ζ := exp 2πi m , then the reflections in I 2 (m) are σ j : (z, z) → zζ j , zζ −j (0 ≤ j < m), and the rotations are ρ j : (z, z) → zζ j , zζ −j .Then σ k σ j σ k = σ 2k−j and ρ −1 k σ j ρ k = σ j+2k ; when m is even, there are two conjugacy classes {σ 2j } and {σ 2j+1 } with 0 ≤ j ≤ m 2 − 1.The real root vector for σ j is v j := sin πj m , − cos πj m and ⟨x, These imply ∆ κ = 4T T .If m is odd, then κ j = κ; if m is even then κ 2j = κ 0 and κ 2j+1 = κ 1 for all j.Denote H v j by H j and let H j = g −1 H j g.In the complex coordinates, Then using (2.1), In R 2 there is only one angular momentum operator (up to scalar multiplication), namely The κ-harmonic polynomials can be found in [4, Section 7.6]; they are expressed in terms of Gegenbauer, respectively Jacobi, polynomials, in case of odd m, respectively even m.

Orthogonal basis of wavefunctions for B 2
Henceforth, we specialize to the group B 2 = I 2 (4).The formulas in the previous section apply with The group has five irreducible representations: four of degree one and one of degree two.The four multiplicative characters satisfy The basis of wavefunctions (solutions of Hψ = 2ω(n + 1 + γ κ )ψ) are denoted ψ n−j,j where the subscript refers to a dominant monomial in the polynomial part (ignoring g) and monomials z a z b (a + b = n) are ordered by |a − b|.The factor g in the wavefunctions will be omitted and we use operators in the form g −1 Ag acting on polynomials.
The basis functions are all expressed in the following: The Jacobi polynomial P (α,β) n (t) of degree n and indices α, β can be defined as (see [4, Proposition 4.14]) this formula leads to the expression stated above.Then define These are of isotype χ 0 , χ 3 , χ 1 , χ 2 , respectively.The L 2 -norms are necessary for normalization, and are derived from The beta function B is defined by a definite integral and satisfies B(a, b) = Γ(a)Γ(b)/Γ(a + b).Denote for polynomials p(z, z) The squared norms are For the odd degrees, p 4n+1 (z) := p 4n,00 (z) + 1 4 p 4n,11 (z) , (4.1) From the orthogonality relations p 4n,00 ⊥ p 4n,11 and p 4n+2,10 ⊥ p 4n+2,01 (different isotypes), Next we list the orthogonal basis, which involves the Laguerre polynomials.The subscript notation may appear strange, but it makes it easy to identify the isotype and every possibility of (n − j, j) can be found by suitably replacing n (the trailing factor g is understood), In this list, σ 0 ψ 2n−j,j = ψ 2n−j,j and σ 0 ψ j,2n−j = −ψ j,2n−j, for 0 ≤ j ≤ n (j < n for the second case).For odd degrees, By construction, Hψ n−j,j = E n ψ n−j,j , where the energy eigenvalue is The squared norms of the ψ follow from the formula where p(z) is homogeneous of degree n and ψ(z) = p(z)L (γκ+n) ℓ (ωzz).When ω = 1, the wavefunctions ψ n−j,j are eigenfunctions of the Dunkl transform with eigenvalue (−i) n (see [4, Theorem 7.7.5]).
5 Some self-adjoint operators

General properties
In this section, we are concerned with finding the action of operators which commute with H on the basis functions described above.Suppose A is such an operator and A is self-adjoint (in L 2 R 2 , h 2 κ dm 2 , then for any (n − j, j) the polynomial Aψ n−j,j is an eigenfunction of H with the same eigenvalue 2ω(n + 1 + γ κ ) and has an expansion n i=0 c i ψ n−i,i (note that dm 2 denotes the R 2 Lebesgue measure and equals rdr dθ for z = re iθ ).Suppose it is known that the topdegree (nonzero) monomials z n−i z i in Aψ n−j,j satisfy m ≤ i ≤ N − m, then ψ n−k,k for k < m or n − m < k ≤ n cannot appear in the expansion of Aψ n−j,j .This is an implicit inductive argument: if z n and z n do not appear, then neither ψ n,0 nor ψ 0,n can appear in the expansion, now consider z n−1 z and zz n−1 , ψ n−1,1 and ψ 1,n−1 and so on.It also follows that it suffices to consider the top-degree terms to find the coefficients of the expansion.The top degree terms of ψ n−j,j are scalar multiples of z n−j z j ± z j z n−j if n is even, and of z n−j z j (or z j z n−j ) if n is odd and n − j > j (or n − j < j).
Definition 5.1.For a polynomial p(z, z), let C(p, z m z n ) denote the coefficient of z m z n in the expansion of p.If p can be expanded in a series of wavefunctions, then C(p, ψ n−j,j ) denotes the coefficient of ψ n−j,j .Suppose, as above, that [A, H] = 0 and A is self-adjoint, then (5.1) generally the coefficients we use are real and the complex conjugate on C can be omitted.

Raising and lowering operators
Formula (3.1) specializes to Proof .We use the polynomial parts H j .First To find the effect of ρ 1 or ρ 3 consider the leading term in ψ m+j,j = p m (z)L (γκ+m) j (ωzz) (total degree n = m + 2j), a scalar multiple of z m+ℓ z ℓ ; examination of each of the formulas for p m shows that each monomial z a z b satisfies a − b = m mod 4, this also applies to ψ j,m+j except for the odd case where the leading term is z j z j+m and a − b = −m mod 4. Also Thus by replacing m by n − 2j, we obtain ρ 1 ψ n−j,j = i n−2j ψ n−j,j ; this applies to all n (if n is even then i n−2j = i 2j−n ).Replace i by −i to find ρ 3 ψ n−j,j .Proposition 5.4.Suppose H 0 ψ n−j,j = n j=0 c j,i ψ n−i,i , then (1) c j,j = 1 2 E n , (2) i = j mod 2 and i ̸ = j implies c i,j = 0, (3) Proof .By hypothesis, From H 0 + H 2 = H, it follows that n i=0 c j,i 1 + (−1) j−i ψ n−i,i = E n ψ n−j,j .Thus 2c j,j = E n and j − i = 0 mod 2 and j ̸ = i implies c j,i = 0. ■ Since {σ 0 , σ 2 } and {σ 1 , σ . This is an element of the center of the group algebra, that is, [R, σ j ] = 0 for 0 ≤ j ≤ 3 and [R, ρ j ] = 0 for 1 ≤ j ≤ 3.
The details are presented in Appendix B. The idea is to use direct (computer-assisted) calculation. ■ The following defines the operator which is the main concern in the sequel; it will be shown to commute with H and the group action but not with angular momentum.The latter claim is proven by demonstrating that eigenfunctions of J 2 are not preserved.commutes with the group action.
Proposition 5.8. Thus this completes the proof.■ 6 The expansion coefficients of H 0

General formulas
This section calculates the coefficients in H 0 ψ n−j,j .Start with To determine the coefficients in H 0 ψ n−j,j = n i=0 c j,i ψ n−i,i , it suffices to consider the monomials of degree n in H 0 ψ n−j,j , that is, analyze (A+B)ψ j,n−j .For a polynomial Let 2j ≤ n and m := n − 2j, then where ε k = (−1) ⌊(k+1)/2⌋ and ⌊r⌋ is the largest integer ≤ r.Thus D Bz n−j z j = 2κ 1 z n−j z j + ε n−2j−1 z j z n−j ; the same formula holds with (z, z) replaced by (z, z) (because [B, σ 0 ] = 0).The special case Bz j z j = 2κ 1 z j z j .
For the odd case, suppose j ≤ n, then This implies

Odd degree
Formula (6.3) is used to find C H 0 ψ 2n+1−j,j , ψ 2n+2−j,j−1 and C H 0 ψ 2n+1−j,j , ψ j,2n+1−j .We will show that the nonzero coefficients in The lower three lines are used to solve for C(H 0 ψ m+j,j , ψ j,m+j ).There are two cases: and second Then 7 The expansion coefficients of K

Even degree
The matrices of H 0 − 1 2 H with respect to the bases {ψ 2n−j,j : 0 ≤ j ≤ n} (σ 0 p = p) and {ψ j,2n−j : 0 ≤ j < n} (σ 0 p = −p) are tridiagonal with zeroes on the main diagonal.If M is such a matrix, then the only nonzero elements in . By use of the expansion coefficients of H 0 − 1 2 H, we find the matrix for 2 (the first three terms act as scalars).

Conclusion
We described an orthogonal basis of wavefunctions in terms of Jacobi and Laguerre polynomials.Each of the basis elements is of a particular isotype, that is, involved in one of the five irreducible representations of the group B 2 .We defined a fourth order differential-difference self-adjoint operator K which commutes with H but not with the angular momentum J 2 .This is an example of superintegrability.The action of K on the basis elements was found explicitly.It is known [9] that there are differential operators of degree 2k which demonstrate superintegrability for the two-parameter I 2 (2k) (even dihedral group) model.It does not appear straightforward to adapt the methods of this paper to the larger groups.

A Transformation of the Hamiltonian
This is a short proof of the formula where h κ (x) := v∈R + |⟨x, v⟩| κv , the W (R)-invariant weight function used in L 2 R N , h 2 κ dm .For the Laplacian, we have The other part of This proves the formula.

B Symbolic computation proofs
There is an analog K(x, y) of the exponential function exp⟨x, y⟩ on R N × R N which satisfies K(x, y) = K(y, x), K(xw, yw) = K(x, y) for all w ∈ W (R) and D This formula together with wK(x, y) = K(xw, y) = K x, yw −1 show how an element of the rational Cherednik algebra (an algebra of operators on polynomials generated by D (x) i , x i : 1 ≤ i ≤ N ∪ W (R)) acts on a generic sum w∈W (R) p W (x, y)K(xw, y).It can be shown that if T is in the rational Cherednik algebra and T K(x, y) = 0, then T = 0 (see Dunkl [3]).For particular groups and operators, the calculation of T K(x, y) can be implemented in computer algebra.
x, y) = y i K(x, y) (where D (x) i is the operator D i acting on x, for 1 ≤ i ≤ N ).The kernel exists for nonsingular parameters {κ v }, which include the situation κ v ≥ 0. Suppose p(x) is a polynomial then by the product ruleD i (p(x)K(x, y)) = y i p(x) + ∂ ∂x i p(x) K(x, y) + v∈R + κ v p(x) − p(xσ v ) ⟨x, v⟩ K(xσ v , y)v i .